Quasi-isometric Classification of Graph Manifold Groups

We show that the fundamental groups of any two closed irreducible nongeometric graph manifolds are quasi-isometric. We also classify the quasi-isometry types of fundamental groups of graph manifolds with boundary in terms of certain finite twocolored graphs. A corollary is the quasi-isometric classification of Artin groups whose presentation graphs are trees. In particular, any two right-angled Artin groups whose presentation graphs are trees of diameter greater than 2 are quasi-isometric; further, this quasi-isometry class does not include any other right-angled Artin groups. A finitely generated group can be considered geometrically when endowed with a word metric. Up to quasi-isometric equivalence, such metrics are unique. (In this article, only finitely generated groups are considered.) Given a collection of groups G, Gromov [11] proposed the fundamental questions of identifying which groups are quasi-isometric to those in G (rigidity) and which groups in G are quasi-isometric to each other (classification). In this article, we focus on the classification question for graph manifold groups and right-angled Artin groups. A compact 3-manifold M is called geometric if M\∂M admits a geometric structure in the sense of Thurston (i.e., a complete locally homogeneous Riemannian metric of finite volume). Thurston’s geometrization conjecture (see [29], [17], [21], [23], [22]) provides that every irreducible 3-manifold of zero Euler characteristic (i.e., with boundary consisting only of tori and Klein bottles) admits a decomposition along tori and Klein bottles into geometric pieces, the minimal such decomposition being called the geometric decomposition. There is a considerable literature on quasi-isometric rigidity and classification of 3-manifold groups. The rigidity results can be briefly summarized in the following form. DUKE MATHEMATICAL JOURNAL Vol. 141, No. 2, c © 2008 Received 18 April 2006. Revision received 29 April 2007. 2000 Mathematics Subject Classification. Primary 20F65; Secondary 57N10, 20F36. Behrstock’s work partially supported by National Science Foundation grant DMS-0604524. Neumann’s work partially supported by National Science Foundation grant DMS-0456227.